Trigonometric Ratios of Allied Angles

Q. The value of $\sin {420}^{\circ }\cos {390}^{\circ }-\cos (-{660}^{\circ })\sin {330}^{\circ }$ is

Q.

If $\sin A=\frac{4}{5}$ and $\cos B=-\frac{12}{13}$, where $A$ and $B$ lie in the first and third quadrant respectively, then $\cos (A+B)=$

1. $\frac{56}{65}$

2. $-\frac{56}{65}$

3. $\frac{16}{65}$

4. $-\frac{16}{65}$

Q. Fundamental period of $f\left(x\right)=|\sin 2x|$ is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{2}$
3. $\pi$
4. $2\pi$

Q.

(a) If $sinA=\frac{12}{13}$ and $sinB=\frac{4}{5},$ where $\frac{\pi }{2}<A<\pi$ and $0<B<\frac{\pi }{2},$ find the following:
(i) sin(A+B)
(ii) cos(A+B)
(b) If sinA= $\frac{3}{5}$, $cosB=\frac{12}{13}$, where A and B both lie in second quadrant, find the value of sin(A+B).

Q.

Prove that: $si{n}^2\frac{\pi }{18}+si{n}^2\frac{\pi }{9}+si{n}^2\frac{7\pi }{18}+si{n}^2\frac{4\pi }{9}=2$

Q.

Prove that:

$\tan \theta .cot\theta =1$

Q. Fundamental period of the function $f\left(x\right)={\sin }^{2}x$ is

1. $\pi$
2. $\frac{\pi }{2}$
3. $2\pi$
4. $4{\pi }^2$

Q.

show that sin 19 ° + sin 41 ° + sin 83 °= sin 23 ° + Sin 37 °+ sin 79 °

Q.

Show that :
(i) $sinAsin(B—C)+sinBsin(C—A)+sinCsin(A—B)=0$
(ii) $sin(B—C)cos(A—D)+sin(C—A)cos(B—D)+sin(A—B)cos(C—D)=0$

Q. The value of $\cos {12}^{\circ }\cos {24}^{\circ }\cos {36}^{\circ }\cos {48}^{\circ }\cos {60}^{\circ }\cos {72}^{\circ }\cos {84}^{\circ }$ is

1. $\frac{1}{64}$
2. $\frac{1}{128}$
3. $\frac{1}{512}$
4. $\frac{1}{256}$

Q. The fundamental period of the function $f\left(x\right)=3+2\sin \left\{\frac{(\pi x+2)}{3}\right\}$ is

Q. If $f\left(x\right)=\frac{\sin \left(2nx\right)}{1+{\cos }^2\left(nx\right)},n\in N$ has $\frac{\pi }{6}$ as its fundamental period, then $n$ is equal to

Q.

If $tanA=\frac{3}{4},cosB=\frac{9}{41},$ where $\pi <A<\frac{3\pi }{2}$ and $0<B<\frac{\pi }{2},$ find $tan(A+B)$.

Q. $f:R\to R$ is a function satisfying the property
$f(2x+3)+f(2x+7)=2\forall x\in R,$ then the fundamental period of $f\left(x\right)$ is

1. $2$
2. $4$
3. $8$
4. $12$

Q. Fundamental period of the function $f\left(\theta \right)=|2\sin 3\theta +4\cos 3\theta |$ is

1. $\frac{2\pi }{3}$
2. $\pi$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{3}$

Q. If $f\left(x\right)=\sin \left(\sqrt{\left[\lambda \right]}x\right)$ is a periodic function with fundamental period $\pi ,$ (where $[.]$ denotes the greatest integer function), then $\lambda$ belongs to

1. $(4,5)$
2. $[4,5)$
3. $(2,3)$
4. $[2,3)$

Q. The value of the expression $\tan \frac{\pi }{7}+2\tan \frac{2\pi }{7}+4\tan \frac{4\pi }{7}+8\cot \frac{8\pi }{7}$ is equal to

1. $\text{cosec}\frac{2\pi }{7}+\cot \frac{2\pi }{7}$
2. $\tan \frac{\pi }{14}-\cot \frac{\pi }{14}$
3. $\frac{\sin \frac{2\pi }{7}}{1-\cos \frac{2\pi }{7}}$
4. $\frac{1+\cos \frac{\pi }{7}+\cos \frac{2\pi }{7}}{\sin \frac{\pi }{7}+\sin \frac{2\pi }{7}}$

Q.

Find the values of the following trigonometric ratios:

$\left(i\right)sin\frac{5\pi }{3}$

$\left(ii\right)sin{3060}^{\circ }$

$\left(iii\right)tan\frac{11\pi }{6}$

$\left(iv\right)cos(-{1125}^{\circ })$

$\left(V\right)tan{315}^{\circ }$

$\left(vi\right)sin{510}^{\circ }$

$\left(vii\right)cos{570}^{\circ }$

$\left(viii\right)sin(-{330}^{\circ })$

$\left(ix\right)cosec(-{1200}^{\circ })$

$\left(X\right)tan(-{585}^{\circ })$

$\left(xi\right)cos{855}^{\circ }$

$\left(xii\right)sin{1845}^{\circ }$

$\left(xiii\right)cos{1755}^{\circ }$

$\left(xiv\right)sin{4530}^{\circ }$

Q.

If $\sin 4A-\cos 2A=\cos 4A-\sin 2A$ (where, $0<A<\frac{\mathrm{\pi }}{4}$) then the value of $\tan 4A$ is:

1. $1$

2. $\frac{1}{\sqrt{3}}$

3. $\sqrt{3}$

4. $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

5. $\frac{\sqrt{3}+1}{\sqrt{3}-1}$

Q.

Prove$\sin (90-A)=\cos A$

Q.

Prove that:

$\left(i\right)tan{720}^{\circ }-cos{270}^{\circ }-sin{150}^{\circ }cos{120}^{\circ }=\frac{1}{4}$

$\left(ii\right)sin{780}^{\circ }sin{480}^{\circ }+cos{120}^{\circ }sin{150}^{\circ }=\frac{1}{2}$

$\left(iii\right)sin{780}^{\circ }sin{120}^{\circ }+cos{240}^{\circ }sin{390}^{\circ }=\frac{1}{2}$

$\left(iv\right)sin{600}^{\circ }cos{390}^{\circ }+cos{480}^{\circ }sin{150}^{\circ }=-1$

$\left(v\right)tan{250}^{\circ }cot{405}^{\circ }+tan{765}^{\circ }cot{675}^{\circ }=0$

Q.

The value of $\tan \left[{\sin }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{3}{\sqrt{13}}\right)\right]$ is

1. $\frac{6}{17}$

2. $\frac{6}{\sqrt{13}}$

3. $\frac{\sqrt{13}}{5}$

4. $\frac{17}{6}$

Q. The absolute value of $\tan \frac{\pi }{16}+\tan \frac{5\pi }{16}+\tan \frac{9\pi }{16}+\tan \frac{13\pi }{16}$ is

Q. The value of $\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{5\pi }{20}\right)\tan \left(\frac{7\pi }{20}\right)\tan \left(\frac{9\pi }{20}\right)$ is

1. $0$
2. $1$
3. $-1$
4. $\infty$

Q. Fundamental period of the function $y=\left|\cot (\frac{x}{4}-\frac{\pi }{3})\right|$ is

1. $\frac{\pi }{3}$
2. $\frac{4\pi }{3}$
3. $2\pi$
4. $4\pi$

Q. If $\theta$ lies in first quadrant and $5\tan \theta =4$, then the value of $\frac{5\sin \theta -3\cos \theta }{\sin \theta +2\cos \theta }$ is

1. $\frac{1}{14}$
2. $\frac{5}{14}$
3. $\frac{3}{14}$
4. $0$
   

Q. $\frac{d}{dx}(3\cos (\frac{\pi }{6}+x^{\circ })-4{\cos }^3(\frac{\pi }{6}+x^{\circ }))=$

1. $\cos \left(3x^{\circ }\right)$
2. $\frac{\pi }{60}\sin \left(3x^{\circ }\right)$
3. $\frac{\pi }{60}\cos \left(3x^{\circ }\right)$
4. $-\frac{\pi }{60}\sin \left(3x^{\circ }\right)$

Q. If $\left\{x\right\}$ denotes the fractional part of $x,$ then value of
$\underset{x\to 0}{lim}\frac{\left\{x\right\}}{\tan \left\{x\right\}}$ is

1. $1$
2. $0$
3. $-1$
4. It does not exist

Q. The value of $\sin {40}^{\circ }{35}^{\prime }\cos {19}^{\circ }{25}^{\prime }+\cos {40}^{\circ }{35}^{\prime }\sin {19}^{\circ }{25}^{\prime }$ is

1. $\frac{\sqrt{3}}{2}$
2. $0$
3. $-1$
4. $\frac{1}{2}$

Q.

Verify that tan30°=tan60°-tan30°/1+tan60°×tan30°